Lipschitz and Holder mappings into jet space Carnot groups

"For $k,n\ge 1$, the jet space J^k(\Rⁿ) is the set of k^th-order Taylor polynomials of functions in C^k(\Rⁿ). Warhurst constructs a Carnot group structure on J^k(\Rⁿ) such that the jets of functions in C^k+1(\Rⁿ) are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\'eodory metric on J^k(\Rⁿ) by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for J^k(\Rⁿ). After describing the Carnot group structure of J^k(\Rⁿ), we will prove that there exists a biLipschitz embedding of \mathbb{S}ⁿ into J^k(\Rⁿ) that does not admit a Lipschitz extension to \mathbb{B}ⁿ⁺¹. This strengthens a result of Rigot and Wenger \cite{RW:LNE} and generalizes a result for \mathbb{H}ⁿ of Dejarnette, Haj{\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\""older equivalence of Carnot groups. We will prove that for all $m\ge 2$ and ε>0, there does not exist an injective, locally (\frac{1}{2}+ε)-H\""older mapping f:\R^m\to J^k(\R) that is locally Lipschitz as a mapping into \R^k+2. This builds on a result of Balogh, Haj{\l}asz, and Wildrick for \mathbb{H}ⁿ. We will conclude by proposing analogues of horizontal and vertical projections for J^k(\R). We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\""assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of J^k(\Rⁿ), focusing primarily on the model filiform jet spaces J^k(\R)."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2019

Author dc:creator

• Jung, Derek

Contributors dc:contributor

• Tyson, Jeremy
• Wu, Jang-Mei
• Fernandes, Rui
• Kaufman, Robert

• Sub-Riemannnian Geometry
• Jet spaces
• Carnot groups
• Geometric Analysis

Statement dc:rights

• Copyright 2018 Derek Jung

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/102425

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/102425